The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case. Introduction.Our purpose is to set a framework for dealing with model uncertainty in mathematical finance and to handle the pricing of contingent claims in this context.Let (S t ) t∈[0,T ] be a real-valued process which stands for an asset price. Usually, it is assumed that the set P m of the equivalent probabilities under which S is a martingale is not empty. This is a sufficient condition to preclude pure gambling strategies that never fail and win with a positive probability.In the classical example of the Black-Scholes model, S solves the linear stochastic differential equation dS t = S t dB t , where

In this paper, we apply change of numeraire techniques to the optimal transport approach for computing model-free prices of derivatives in a two-period setting. In particular, we consider the optimal transport plan constructed in Hobson and Klimmek (Finance Stoch. 19:189-214, 2015) as well as the one introduced in Beiglböck and Juillet (Ann. Probab. 44:42-106, 2016) and further studied in HenryLabordère and Touzi (Finance Stoch. 20:635-668, 2016). We show that in the case of positive martingales, a suitable change of numeraire applied to Hobson and Klimmek (Finance Stoch. 19:189-214, 2015) exchanges forward start straddles of type I and type II, so that the optimal transport plan in the subhedging problems is the same for both types of options. Moreover, for Henry-Labordère and Touzi's (Finance Stoch. 20:635-668, 2016) construction, the right-monotone transference plan can be viewed as a mirror coupling of its left counterpart under the change of numeraire.Keywords Robust hedging · Model-independent pricing · Model uncertainty · Optimal transport · Change of numeraire · Forward start straddle Mathematics Subject Classification (2010) 91G20 · 91G80 JEL Classification G13

Abstract. The rough Bergomi model introduced by Bayer, Friz and Gatheral [3] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen, Lunde and Pakkanen [4].

The aim of this work is to introduce a new stochastic volatility model for equity derivatives. To overcome some of the well-known problems of the Heston model, and more generally of the affine models, we define a new specification for the dynamics of the stock and its volatility. Within this framework we develop all the key elements to perform the pricing of vanilla European options as well as of volatility derivatives. We clarify the conditions under which the stock price is a martingale and illustrate how the model can be implemented.

The rough Bergomi model introduced by Bayer, Friz and Gatheral [3] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen, Lunde and Pakkanen [4]. IntroductionVolatility, though not directly observed nor traded, is a fundamental object on financial markets, and has been the centre of attention of decades of theoretical and practical research, both to estimate it and to use it for trading purposes. The former goal has usually been carried out under the historical measure (P) while the latter, through the introduction of volatility derivatives (VIX and related family), has been evolving under the pricing measure Q. Most models used for pricing purposes (Heston [19], SABR [17], Bergomi [5]) are constructed under Q and are of Markovian nature (making pricing, and hence calibration, easier). Recently, Gatheral, Jaisson and Rosenbaum [14] broke this routine and introduced a fractional Brownian motion as driving factorof the volatility process. This approach (Rough Fractional Stochastic Volatility, RFSV for short) opens the door to revisiting classical pricing and calibration conundrums. They, together with the subsequent paper by Bayer, Friz and Gatheral, (see also [1,12]) in particular showed that these models were able to capture the extra steepness of the implied volatility smile in Equity markets for short maturities, which continuous Markovian stochastic volatility models fail to describe. The icing on the cake is the (at last!!) reconciliation between the two measures P and Q within a given model, showing remarkable results both for estimation and for prediction.One of the key issues in Equity markets is, not only to fit the (SPX) implied volatility smile, but to do so jointly with a calibration of the VIX (Futures and ideally options). Gatheral's [15] double mean reverting process is the leading (Markovian) continuous model in this direction, while models with jumps have been proposed abundantly by Carr and Madan [9] and Kokholm and Stisen [20]. This issue was briefly tackled by Bayer, Friz and Gatheral [3] for a particular rough model (rough Bergomi), and we aim here at providing a deeper analysis of VIX dynamics under this rough model and at implementing pricing schemes for VIX Futures and options.Our main contribution is a precise link between the forward variance curve (ξ T (·)) T ≥0 and the initial forward variance curve ξ 0 (·) in the rough Bergomi model. This in turn, allows us not only to provide simulation methods Date: January 17, 2017.2010 Mathematics Subject Classification. 91G20, 91G99, 91G60, 91B25.